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Bornivorous Set
In functional analysis, a subset of a real or complex vector space X that has an associated vector bornology \mathcal is called bornivorous and a bornivore if it absorbs every element of \mathcal. If X is a topological vector space (TVS) then a subset S of X is bornivorous if it is bornivorous with respect to the von-Neumann bornology of X. Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces. Definitions If X is a TVS then a subset S of X is called and a if S absorbs every bounded subset of X. An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets). Infrabornivorous sets and infrabounded maps A linear map between two TVSs is called if it maps Banach disks to bounded disks. A disk in X is called if it absorbs every Banach disk. An absorbing disk in a locally conve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bornology
In mathematics, especially functional analysis, a bornology \mathcal on a vector space X over a field \mathbb, where \mathbb has a bornology ℬ\mathbb, is called a vector bornology if \mathcal makes the vector space operations into bounded maps. Definitions Prerequisits A on a set X is a collection \mathcal of subsets of X that satisfy all the following conditions: #\mathcal covers X; that is, X = \cup \mathcal #\mathcal is stable under inclusions; that is, if B \in \mathcal and A \subseteq B, then A \in \mathcal #\mathcal is stable under finite unions; that is, if B_1, \ldots, B_n \in \mathcal then B_1 \cup \cdots \cup B_n \in \mathcal Elements of the collection \mathcal are called or simply if \mathcal is understood. The pair (X, \mathcal) is called a or a . A or of a bornology \mathcal is a subset \mathcal_0 of \mathcal such that each element of \mathcal is a subset of some element of \mathcal_0. Given a collection \mathcal of subsets of X, the smallest bornolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absorbing Set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset. Definition Notation for scalars Suppose that X is a vector space over the field \mathbb of real numbers \R or complex numbers \Complex, and for any -\infty \leq r \leq \infty, let B_r = \ \quad \text \quad B_ = \ denote the ''open ball'' (respectively, the ''closed ball'') of radius r in \mathbb centered at 0. Define the product of a set K \subseteq \mathbb of scalars with a set A of vectors as K A = \, and define the product of K \subseteq \mathbb with a single vector x as K x = \. Preliminaries Balanced core and balanced hull A subset S of X is said to be '' '' if a s \in S for all s \in S and all scalars a satisfying , a, \leq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bornological Space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after , the French word for " bounded". Bornologies and bounded maps A on a set X is a collection \mathcal of subsets of X that satisfy all the following conditions: \mathcal covers X; that is, X = \cup \mathcal; \mathcal is stable under inclusions; that is, if B \in \mathcal and A \subseteq B, then A \in \mathcal; \mathcal is stable under finite union ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Set (topological Vector Space)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be ''inflated'' to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935. Definition Suppose X is a topological vector space (TVS) over a topological field \mathbb. A subset B of X is called or just in X if any of the following equivalent conditions are satisfied: : For every neighborhood V of the origin there exists a real r > 0 such that B \subseteq s VFor any set A and scalar s, the notation s A denotes the set s A := \. for all scalars s satisfying , s, \geq r. * This was the definition introduced by John ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex topological vector spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, then the or of K is defined to be the function p_K : X \to , \infty valued in the extended real numbers, defined by p_K(x) := \inf \ \quad \text x \in X, where the infimum of the empty set is defined to be positive infinity \,\infty\, (which is a real number so that p_K(x) would then be real-valued). The set K is often assumed/picked to have properties, such as being an absorbing disk in X, that guarantee that p_K will be a real-valued seminorm on X. In fact, every seminorm p on X is equal to the Minkowski functional (that is, p = p_K) of any subset K of X satisfying \ \subseteq K \subseteq \ (where all three of these sets are necessarily absorbing in X and the first and last are also disks). Thus every seminorm (which is a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auxiliary Normed Space
In functional analysis, a branch of mathematics, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk D is bounded: in this case, the auxiliary normed space is \operatorname D with norm p_D(x) := \inf_ r. The other method is used if the disk D is absorbing: in this case, the auxiliary normed space is the quotient space X / p_D^(0). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces). Induced by a bounded disk – Banach disks Throughout this article, X will be a real or complex vector space (not necessarily a TVS, yet) and D will be a disk in X. Seminormed space induced by a disk Let X will be a real or complex vector space. For any subset D of X, the ''Minkowski functional'' of D defined by: *If D = \varnothing then define p_(x) : \ \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Disk
In functional analysis, a branch of mathematics, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk D is bounded: in this case, the auxiliary normed space is \operatorname D with norm p_D(x) := \inf_ r. The other method is used if the disk D is absorbing: in this case, the auxiliary normed space is the quotient space X / p_D^(0). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces). Induced by a bounded disk – Banach disks Throughout this article, X will be a real or complex vector space (not necessarily a TVS, yet) and D will be a disk in X. Seminormed space induced by a disk Let X will be a real or complex vector space. For any subset D of X, the ''Minkowski functional'' of D defined by: *If D = \varnothing then define p_(x) : \ \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudometrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS. Pseudometrics and metrics A pseudometric on a set X is a map d : X \times X \rarr \R satisfying the following properties: d(x, x) = 0 \text x \in X; Symmetry: d(x, y) = d(y, x) \text x, y \in X; Subadditivity: d(x, z) \leq d(x, y) + d(y, z) \text x, y, z \in X. A pseudometric is called a metric if it satisfies: Identity of indiscernibles: for all x, y \in X, if d(x, y) = 0 then x = y. Ultrapseudometric A pseudometric d on X is called a ultrapseudometric or a strong pseudometric if it satisfies: Strong/Ultrametric triangle inequality: d(x, z) \leq \max \ \text x, y, z \in X. Pseudometric space A pseudometric space is a pair (X, d) consisting of a set X and a pseudometric d on X such that X's to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
